Theorem prnz 4741

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4726	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4307	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ∅c0 4296  {cpr 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-pr 4592

This theorem is referenced by:  opnz  5433  propssopi  5468  fiint  9277  fiintOLD  9278  wilthlem2  26979  upgrbi  29020  wlkvtxiedg  29553  shincli  31291  chincli  31389  constrextdg2lem  33738  spr0nelg  47477  sprvalpwn0  47484